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Modern Finance > Week 9- Options > Flashcards

Week 9- Options Flashcards

1
Q

What does the value of a derivative come from?

A

The value of a derivative is derived from the value of the underlying asset.

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2
Q

What are options?

A

Options are special type of financial asset which give the holder the right but not the obligation to buy/sell a particular security at a predetermined price.

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3
Q

What do options allow?

A

The hedging against risk

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4
Q

When writing an option, what does this do for the risk?

A

Transfers it from you to a different part

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5
Q

What is the right to buy called?

A

A call option

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6
Q

What is the right to sell called?

A

A put option

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7
Q

What is the price at which an option can be exercised called?

A

The exercise/strike price

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8
Q

The option of whether to buy or sell has a value. What does the buyer of an option pay for this privilege?

A

A premium

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9
Q

If it is profitable to exercise the option at the asset’s current price, then it is the..

A

The money option

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10
Q

If it is unprofitable to exercise the option at the asset’s current price, then it is the..

A

the out of money option

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11
Q

If an option ca only be exercised on a particular day, what is this called?

A

A European Call

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12
Q

If an option can be exercised on or before a specific date, what is this called?

A

An American Call

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13
Q

Are premiums higher for American or European calls?

A

American, as you get more flexibility.

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14
Q

How do we calculate the value of a call option?

A

Value of the call option CT with stock price of ST and exercise price X is:
CT = max[0,ST – X]
- max refers to the maximum between (ST -X) and 0. (ie the contract value cannot be less than 0, as 0 is worthless)

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15
Q

In which case is the call worth the difference between the current stock price and the exercise price.

A
  • The call is worth the difference between the current stock price and the exercise price. π depends upon whether CT >a : a is the
    premium.
  • If ST <X then -π given +a (then call option has no value and is not exercised)
  • If ST >X and a< CT then +π (then call in the money at termination)
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16
Q

In which case does exercising a sell call turn a profit?

A

If S-X-a>0 (ie C-a>0)
where S = share price
X= exercise price
a= premium

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17
Q

Give one key difference between a short and an option.

A

In a short you have an obligation to repay, in options there are no obligations.

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18
Q

What is the payoff from buying a call option?

A

a-C < 0 is where profit is made. This is the mirror image from buying a call option.

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19
Q

With a call option, when will a rational investor exercise? When they make profit?

A

No, but when the payoff is greater than a premium (ie potentially between the negative payoff of the premium and 0)

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20
Q

How do we calculate the value of a put option?

A

Value of the put option CT with stock price of ST and exercise price X is:
CT = max[0,X- ST]

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21
Q

What is a put worth, and when is it in profit?

A
  • The put is worth the difference between the current stock price and the exercise price. π depends upon whether PT >b where b is the
    premium.
  • If X> ST and b<P then +π .
  • If X<ST then -π given +b.
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22
Q

When would the value of a put contract be zero?

A

If the stock price is higher than the specified exercise/strike price, as naturally you would rather sell for that instead.

23
Q

Why are we dealing with European auctions instead of American?

A

As they are easier to value as there is only a specific date of termination, unlike an American one where there is flexibility.

24
Q

What are interest rate options used for?

A

To give the holder a right (but not obligation), to lend or borrow a stated amount at a predetermined rate.

25
Q

What are currency options used for?

A

To give the holder the right, but not the obligation to buy or sell a certain amount of currency at a given exchange rate.

26
Q

What are options also commonly traded by?

A

Speculators purely for profit.

27
Q

Does adding T-Bills to a portfolio on the Capital Market Line alter the shape of the distribution?

A

No, it merely squeezes te distribution as it lowers the variance.

28
Q

Can options modify the shape of the return distribution?

A

Yes

29
Q

What does combining a portfolio with a put do?

A

Eliminates risk below r*, it effectively eliminates negative outcomes

30
Q

An option premium consists of which two components?

A

Intrinsic value and time value

31
Q

What is the intrinsic value for a call option?

A

Cash price - Strike price

32
Q

What is the intrinsic value for a put option?

A

Strike price - cash price

33
Q

Describe in/out/at the-money for call options.

A
  • If an intrinsic value for an option exists then in-the-money.
  • If the strike price is above the cash price for a call option then it has zero intrinsic value and is out-of-the-money.
  • If the strike price=cash price then at-the-money.
34
Q

What does the intrinsic value reflect the price of?

A

The intrinsic value reflects the price that could be received if the option were “locked in” today at the current market price.

35
Q

What does time value equal, if option premium = time value + intrinsic value? Also describe what it reflects.

A
  • Time value = option premium – intrinsic value
  • Time value reflects the fact that an option may have more ultimate value than the intrinsic value alone.
  • Hence, even if option is out-of-the-money a buyer can hope that at some time prior to expiration changes in the spot price will
    move the option into the money.
36
Q

Why could time value lead to an out-of-money option being in money?

A

As the price of the stock may change (ie volatility)

37
Q

What is the upper bound on the European call option?

A

Upper bound: The value of a call option Ct is below the value of the stock St for all t ≤ T
Ct < St

38
Q

What is the lower bound on the European call option?

A

Lower bound: The value of a call option Ct is at least the value of the stock St less the discounted value of the cost of exercising the
call (r is risk-free rate of interest) for all t < T
Ct > St – Xe^-r(T – t)

39
Q

What iss Xe^-r(T – t)?

A

The present value of X upon continuoiuus discounting

40
Q

To understand the upper and lower bounds, we need what?

A

The no-arbitrage principle- ie an investor cannot make money out of nothing. A portfolio that makes a positive return in all states of the world (return with no downside risk below zero) must cost something to construct.

41
Q

What is the no-arbitrage principle also sometimes known as?

A

The “no magic money pump” argument.

42
Q

Explain why the upper bound is Ct<St

A
  • Time t < T: Suppose Ct ≥ St. Investor A sells a call option on the stock to B and uses some of the proceeds to buy the stock. So, investor A makes a sure gain Ct – St ≥ 0 (constructing this portfolio costs the investor less than nothing).
  • Time T: If investor B exercises the call option, A sells the stock to investor B, making X. If investor B does not exercise the call option, A sells the stock to investor C, making ST. Either way, it’s a sure gain of at least min[X, ST] > 0 to investor A.
  • Conclusion: Ct ≥ St leads to a money pump. So, Ct < St
43
Q

Explain why the lower bound for all t < T is
Ct > St – Xe^-r(T – t)

A
  • Time t < T: Investor simultaneously (i) short-sells the stock; (ii)buys a call option in the stock; and (iii) buys Xe^-r(T – t) of the riskfree asset. The value of the resulting portfolio is
    Vt = Ct – St + Xe-r(T – t)
  • Time T: If investor exercises the call option (ST > X) it is worth ST– X. The risk-free assets have earned interest and are now worth X. The short-sold stock will now cost ST to buy back. Value of the portfolio if the call is exercised is therefore now
    VT = ST – X – ST + X = 0, but if the option has no value is Vt= -St+X≥ 0
  • Either way, the portfolio always has a (weakly) positive value.
    If Vt ≤ 0 then the portfolio costs less than nothing to construct, but yields a guaranteed (weakly) positive return. To avoid the money pump problem, the original portfolio must
    have cost money to construct (Vt > 0). So
    Ct > St – Xe-r(T – t)
44
Q

What are the 6 underlying assumptions for the Black and Scholes formula to evaluate European options?

A
  • Underlying asset pays no dividends or interest during lifetime;
  • Option is European, i.e. cannot be exercised prior to maturity;
  • Risk free rate is fixed over the lifetime of the asset;
  • Financial markets are perfect, i.e. no transaction costs or taxes;
  • Price of underlying asset is log normally distributed; (ie there are no negative values)
  • Price moves in continuous time, assuming that stock prices will only move slightly ↑↓ during the next microsecond.
45
Q

Is the value of St – Xe^-r(T – t) at time T certain?

A

No, look at the expected value upon expiration.

46
Q

What is the Black-Scholes model formula?

A

Ct = StN(d1) – Xe^-r(T – t) N(d2)
where:
-𝐶(𝑡) is the call value of the option with t left before expiration;
- 𝑡 is time until the option expires;
- 𝑆 is current/spot stock price
- 𝑋 is exercise (strike) price of the option
- 𝑒 is 2.71828 – base of the natural system of logs
- 𝑟 is the continuously compounded risk free rate of interest
- 𝑁(𝑑𝑖) is the value of the cumulative normal density function

47
Q

What does the Black-Scholes state that the cakk value is effectively equal to? What does this mean for each term?

A

Call Value = (share price×probability1) –
(present value of exercise price×probability2)
* Thus the first term is the expected stock (option) price at period t
* The second term is the expected present value of the exercise price at t.
* The value of the call option is the difference between the two terms

48
Q

Do increases these variables increase or decrease the call option value?
Price volatility
Time to expiration
Exercise price
Current stock price
Risk free interest rate

A

Price volatility ↑
Time to expiration ↑
Exercise price ↓
Current stock price ↑
Risk free interest rate ↑

49
Q

So as the valu of the underlying stock rises, what happens to the intrinsic value of the call option?

A

It rises

50
Q

Why does it usually make more sense to sell the option before it expires?

A

As it should also have time value,as well as just its intrinsic value.

51
Q

What does an increase in risk do to the value of the option?

A

It increases it

52
Q

When does the 45-degree line meet the horizontal axis and what does it represent?

A
  • The 45 degree line meets the horizontal axis at the PV of the exercise price X discounted at the risk-free rate.
  • The 45 degree line represents the difference between the share price S and the present value of the exercise price X.
53
Q

Give 3 practical issues with Black-Scholes

A
  • Quantities such as the stock variance and the risk-free rate assumed to be constant during the life of the option, but in practice these vary
  • Large price changes are more frequently observed in the real world than those expected and implied by the Black-Scholes
    model
  • Lognormal prices are a questionable assumption for most stocks

Modern Finance (9 decks)

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